Lemma 36 [1]}  Let \(X_1,X_2,\cdots ,X_n\)  be a stationary Markov chain on some Polish space \(\mathcal {S}\)  with pseudo spectral gap \(\gamma _{\rm {ps}}\) , with \(|f(x)-\mathbb {E}_{\pi }(f)|\le C\)  for every \(x \in \mathcal {S}\) . Let the projection operator \(\mathbf {\pi }: L_2(\pi )\rightarrow L_2(\pi )\)  be defined as \(\mathbf {\pi }(f)(x)=\pi (f)=\int _{\mathcal {S}} f d\pi \) . Define \(V_f:=\operatorname{\mathsf {Var}}_{\pi }(f)\le C^2\)  and \(S_n:=\sum _{i=1}^n f(X_i)\) . Then, for any \(u>0\) , the following holds:
\(\mathbb {P}\bigg [\frac{1}{n}\bigg |S_n-\mathbb {E}_{\pi }(S_n)\bigg |\ge u \bigg ] &\le 2 \exp \bigg (-\frac{n^2 u^2 \gamma _{\rm {ps}}}{8(n+1/\gamma _{\rm {ps}})V_f+20 nu C} \bigg )\\&\le 2 \exp \bigg (-\frac{n u^2 \gamma _{\rm {ps}}}{8(1+1/\gamma _{\rm {ps}})C^2+20 u C} \bigg ).\) 
